Force

In classical physics, force is defined as the product of a body's mass and acceleration. In other words, how hard you push on something determines how rapidly you change its speed. Of course, the heavier it is, the more it resists this change. So, what you're really changing is the body's momentum.

Using advanced mathematics, force may be defined as the time rate of change of momentum of a body $\vec F = {d \vec p \over dt}$. The SI unit of force is the newton and the US customary system unit is the pound.

Classically, the momentum of an object is given by $\vec p = m \vec v$ and acceleration relates to force via Newton's Second Law as $\vec F = m \vec a$ when mass can be assumed to be constant. More generally it is prescribed as $\vec F = \frac{d \vec p}{dt}$.[1] In these expressions, F stands for the total vector sum of all forces, m for the mass of the object, a for its acceleration expressed as a vector, p stands for momentum vector and v for velocity vector. In special relativity, these presciptions must be modified so as to be Lorentz invariant, which among other things, means that all inertial reference frames stand on equal footing and have the same prescription for all physical and dynamical quantities, though observers in different inertial reference frames will measure different values for many of them, each observer being correct for his own frame(!).

There are four known fundamental types of forces occurring in nature[2]:

Electromagnetic force
The force that acts on objects with electric charge.
Gravitational force
The force that attracts any two objects with mass.
Strong force
The force that keeps atomic nuclei together.
Weak force
This force is involved in beta decay, in which a neutron in an atomic nucleus is changed to a proton, emitting an electron and a neutrino.

It should be noted that the latter two forces have an extremely short range (on the order of femtometers), and that a classical (Newtonian or relativistic) description of these forces is not possible. They can only be described using quantum field theory, a relativistic version of quantum mechanics.

References

1. Marcelo Alonso and Edward J. Finn, Fundamental University Physics, Addison-Wesley.
2. Lewis H. Ryder, Quantum Field Theory, 2nd ed., Cambridge University Press, Cambridge (UK), 1996